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Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral

Salvador Sánchez-Perales, Francisco J. Mendoza-Torres (2020)

Czechoslovak Mathematical Journal

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation - y ' ' + q y = f , where q and f are Henstock-Kurzweil integrable functions on [ a , b ] . Results presented in this article are generalizations of the classical results for the Lebesgue integral.

Bounded linear functionals on the space of Henstock-Kurzweil integrable functions

Tuo-Yeong Lee (2009)

Czechoslovak Mathematical Journal

Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.

Boundedness properties of fractional integral operators associated to non-doubling measures

José García-Cuerva, A. Eduardo Gatto (2004)

Studia Mathematica

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications

Silvestru Sever Dragomir (2014)

Communications in Mathematics

Some new bounds for the Čebyšev functional in terms of the Lebesgue norms f - 1 b - a a b f ( t ) d t [ a , b ] , p and the Δ -seminorms f p Δ : = a b a b | f ( t ) - f ( s ) | p d t d s 1 p are established. Applications for mid-point and trapezoid inequalities are provided as well.

Bounds for f -divergences under likelihood ratio constraints

Sever Silvestru Dragomir (2003)

Applications of Mathematics

In this paper we establish an upper and a lower bound for the f -divergence of two discrete random variables under likelihood ratio constraints in terms of the Kullback-Leibler distance. Some particular cases for Hellinger and triangular discimination, χ 2 -distance and Rényi’s divergences, etc. are also considered.

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

Bounds for sine and cosine via eigenvalue estimation

Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski, Alexander Kovacec (2014)

Special Matrices

Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, [...] Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain...

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